Evaluate the following integral:
$\int \frac{1}{\left(1+x^{2}\right) \sqrt{1-x^{2}}} d x$
assume $\mathrm{x}=\frac{1}{\mathrm{t}}$
$d x=-\frac{1}{t^{2}} d t$
$-\int \frac{t d t}{\left(t^{2}+1\right)\left(\sqrt{t^{2}-1}\right.}$
Let $\mathrm{t}^{2}-1=\mathrm{u}^{2}$'
$\mathrm{tdt}=\mathrm{udu}$
$-\int \frac{u d u}{\left(u^{2}+2\right) u}$
$-\int \frac{d u}{\left(u^{2}+2\right)}$
Using identity $\int \frac{1}{x^{2}+1} d x=\arctan (x)$
$-\frac{1}{\sqrt{2}} \arctan \left(\frac{\mathrm{u}}{\sqrt{2}}\right)+\mathrm{c}$
Substituting $u=\sqrt{t^{2}-1}$
$-\frac{1}{\sqrt{2}} \arctan \left(\frac{\sqrt{t^{2}-1}}{\sqrt{2}}\right)+c$
Substituting $\mathrm{t}=\frac{1}{\mathrm{x}}$
$-\frac{1}{\sqrt{2}} \arctan \left(\frac{\sqrt{\frac{1}{x^{2}}-1}}{\sqrt{2}}\right)+c$
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